A Catholic Reflection on Nicomachus’ Arithmetic, Book I, Chapter 8
In the eighth chapter of Arithmetic, Nicomachus brings our attention to one of the most orderly and structured types of numbers: what he calls the even-times-even. He doesn’t simply list them or explain how to calculate them. Instead, he reflects on how their structure reveals a kind of mathematical harmony—one that mirrors the wisdom and order God placed into creation itself.
For us, as Catholic readers, this is not just a study of arithmetic—it is a chance to contemplate the mind of God, who, as Sacred Scripture tells us, “has arranged all things by measure and number and weight” (Wisdom 11:20). As we work through this lesson, we’ll explore how Nicomachus helps us see number not only as a tool, but as a symbol of divine truth, balance, and purpose.
Every Number Is Surrounded by Others—and Reflects Them
Nicomachus begins with a simple but beautiful observation. Every number, he says, is the average of the two numbers on either side of it. In other words, if you take any number, it is halfway between the number just below it and the number just above it.
Take 5, for example. The number before it is 4, the number after it is 6. Add 4 and 6, and you get 10. Half of that is 5. This is always true—and not just with the closest neighbors. A number is also halfway between the two numbers that are two steps away (like 3 and 7), or three steps away (like 2 and 8), and so on, as far as you like.
The only exception, Nicomachus says, is the number one. Because one is the very first number, it doesn’t have a number before it to create a pair. Instead, it is simply half of the next number—two. That makes one the natural starting point of all number. It is the seed from which all numerical relationships grow.
This reminds us of a truth in our faith as well. God is One—absolute, eternal, and without equal. From His unity flows all multiplicity, all structure, all creation. In the same way, from the unit (one) comes all number, all proportion, and all harmony.
Three Types of Even Numbers
Nicomachus now turns to a deeper topic: how even numbers can be divided and classified. He explains that even numbers can be grouped into three kinds:
Even-times-even
Odd-times-even
Even-times-odd
These names might seem complicated at first, but let’s look at what they mean.
An even-times-even number is one that can be divided by two again and again, over and over, until we reach one, which cannot be divided any further. It is “even all the way down.”
An even-times-odd number is an even number, but when you divide it by two just once, you’re left with an odd number.
An odd-times-even number is the one that sits in between—it shares some traits of each. It’s even, but its divisions mix even and odd.
So, Nicomachus places these three in a kind of order. The even-times-even and even-times-odd are opposites—like the two ends of a scale. The odd-times-even is in the middle, acting like a bridge between the other two.
What Makes a Number Even-Times-Even?
Let’s look more closely at the even-times-even numbers.
These are the numbers that you can keep dividing by two repeatedly, and at every step, you still get a number that’s also even—until you finally reach one.
Take 64 as an example. Here’s how it breaks down:
Half of 64 is 32.
Half of 32 is 16.
Half of 16 is 8.
Half of 8 is 4.
Half of 4 is 2.
Half of 2 is 1.
At every step until the very last, each result is still an even number. This kind of structure shows a complete internal harmony—it’s like a ladder built entirely from even steps.
Nicomachus notes that this is the key trait of even-times-even numbers: no matter which part of the number you look at—even a small part—it will always reflect the nature of the whole. Every piece is itself an even-times-even number.
This is why Nicomachus says that even-times-even numbers are pure in both name and value. They don’t change when you break them apart. They stay true to their nature, from the largest parts to the smallest.
How to Create All Even-Times-Even Numbers
Nicomachus then gives us a method to generate every even-times-even number. It’s beautifully simple:
Start with one, and keep multiplying by two.
This process creates a series like this:
1, 2, 4, 8, 16, 32, 64, 128, 256, 512…
Each of these numbers is formed by doubling the one before it, and all of them are even-times-even (except for 1, which is the unit).
What’s more, every part of any number in this sequence is also in the sequence. If you take a part of 64, for example, like 16 or 8, you’ll find those numbers earlier in the series. Each part has a matching role, both in name and in value. The structure of the whole series is carefully balanced.
Correspondence and Harmony in the Series
Nicomachus observes something very elegant. The numbers in this series are not just random values—they are arranged in perfect relationships of correspondence.
If the series has an even number of terms, there will be two middle terms, and the parts will correspond in pairs:
The numbers near the center are related to each other as factors.
The numbers further from the center still correspond, but in inverse proportions.
Take the example of 1 to 128, which has eight terms. The middle terms are 8 and 16. These two are related in this way:
16 is one-eighth of 128.
8 is one-sixteenth of 128.
The pairs around them follow the same kind of logic:
32 is one-fourth of 128, and 4 is one-thirty-second.
64 is one-half, and 2 is one-sixty-fourth.
And finally, 1 is one-one-hundred-twenty-eighth of the whole, and 128 is the full number compared to 1.
It’s like looking at a mirror: every number has a corresponding reflection, and the whole structure is beautifully balanced.
If the series has an odd number of terms, like from 1 to 64, there is a single middle term, and it corresponds to itself. All the other terms still reflect each other symmetrically around that center point.
A Pattern of Summation and the Nearness of Perfection
Nicomachus notes another interesting property: if you add up all the numbers in the series before a given term, the sum will be just one unit less than that term.
For example:
1 + 2 + 4 + 8 + 16 + 32 = 63, which is one less than 64.
1 + 2 + 4 + 8 + 16 + 32 + 64 = 127, which is one less than 128.
This might seem like a small detail, but it reveals something powerful: the numbers in this series are always just shy of becoming the next one. They hint at completion, but fall short by just one unit. This will become important in Nicomachus’ later teaching on perfect numbers, where this kind of nearness to balance has deeper meaning.
It also invites a spiritual reflection. Just as the sum of these numbers falls short by one, human nature, on its own, always falls just short of perfection. Only by the grace of God is our nature completed and made whole.
A Law of Proportions: Products of Extremes and Means
Finally, Nicomachus gives us a mathematical principle that applies to these series:
If the series has an even number of terms, the product of the extreme terms (the first and last) equals the product of the middle pairs.
If the series has an odd number of terms, the product of the extremes equals the square of the middle term.
So:
In 1 to 128 (even number of terms):
1 × 128 = 2 × 64 = 4 × 32 = 8 × 16In 1 to 64 (odd number of terms):
1 × 64 = 2 × 32 = 4 × 16 = 8 × 8
This mathematical law reflects a hidden order and unity behind the structure of these numbers. And for the philosopher, this harmony is more than arithmetic—it is a sign that the world is made with wisdom, not chance.
Conclusion: A World Built on Order
In this lesson, Nicomachus doesn’t just teach us about even-times-even numbers. He shows us how deeply ordered creation really is. Starting from one, through simple doubling, a whole universe of structured relationships appears—each number connected, each part reflecting the whole.
For Catholic readers, this is not just mathematical elegance. It is the beauty of divine providence. God made all things with number and proportion, not to confuse us, but to reveal His glory through order. In every well-structured number, we see a glimpse of the divine reason that governs the heavens.
And so, as we study arithmetic, we are not just learning about numbers—we are drawing near to the wisdom that made the world.
Mr. William C. Michael, O.P.
Headmaster
Classical Liberal Arts Academy
References
Aristotle. Nicomachean Ethics. Translated by W. D. Ross. Perseus Digital Library. https://www.perseus.tufts.edu/hopper/text?doc=Aristot.+Nic.+Eth.
Catechism of the Catholic Church. 2nd ed. Vatican: Libreria Editrice Vaticana, 1997. https://www.vatican.va/archive/ENG0015/_INDEX.HTM
Holy Bible. New American Bible, Revised Edition (NABRE). United States Conference of Catholic Bishops, 2011. https://bible.usccb.org/
Nicomachus of Gerasa. Introduction to Arithmetic. Translated by Martin L. D’Ooge. New York: Macmillan, 1926. https://archive.org/details/NicomachusIntroToArithmetic
Plato. Timaeus. Translated by Benjamin Jowett. Perseus Digital Library. https://www.perseus.tufts.edu/hopper/text?doc=Plat.+Tim.
St. Augustine. On Music (De Musica). In The Fathers of the Church, Vol. 2. Washington, D.C.: Catholic University of America Press. Also available at: https://www.newadvent.org/fathers/
St. Thomas Aquinas. Summa Theologica. Translated by the Fathers of the English Dominican Province. New York: Benziger Bros., 1947. https://www.newadvent.org/summa/
Wisdom 11:20. New American Bible. https://bible.usccb.org/bible/wisdom/11